Q. 22 - Calculus

Question

Express the signed area between the graph of $y=\frac{1}{x}$and the $x$-axis from $x=0.25$ to $x=1$ in terms of logarithms.

Step-by-Step Solution

Answer

The signed area between $y=\frac{1}{x}$ from $x=0.25$ to $x=1$ in terms of logarithms is, $-\ln \left(0.25\right)$.

Step 1 . Given information

 $y=\frac{1}{x}$ and the $x$-axis from $x=0.25$ to $x=1$.

Step 2 . Find the signed area.

$$\begin{gathered} {\int }_{0.25}^1\frac{1}{x}dx=[\ln x{]}_{0.25}^1 \\ =\ln 1-\ln (0.25) \\ =0-\ln (0.25) \\ =-\ln (0.25) \end{gathered}$$

Therefore, the signed area is, $-\ln \left(0.25\right)$.